SPACECRAFT TRAJECTORY DESIGN TECHNIQUES USING RESONANT ORBITS A Thesis Submitted to the Faculty of Purdue University by Srianish Vutukuri In Partial Fulfillment of the Requirements for the Degree of Master of Science in Aeronautics and Astronautics May 2018 Purdue University West Lafayette, Indiana ii THE PURDUE UNIVERSITY GRADUATE SCHOOL STATEMENT OF THESIS APPROVAL Professor Kathleen C. Howell, Chair School of Aeronautics and Astronautics Professor Carolin E. Frueh School of Aeronautics and Astronautics Professor David A. Spencer School of Aeronautics and Astronautics Approved by: Professor Weinong Wayne Chen Aeronautics and Astronautics Associate Head for Graduate Education iii Dedicated to Mom, Dad and Amarsh. iv ACKNOWLEDGMENTS I would like to thank my advisor first, Professor Kathleen Howell for giving me this fantastic opportunity to be a part of the Multi-Body Dynamics research group. You have always encouraged me to pursue the hard things and at the same time been my constant guiding force. You are an inspiration to me both personally and professionally. I would like to extend my thanks to committee members, Professor David Spencer and Professor Carolin Frueh, for their valuable advice and feedback. A big thank you to Professor Micheal Grant, Professor James Longuski and Professor Sarag Saikia for continually motivating me in your classes. My heartfelt thank you to all the group mates for your constant care and support throughout my time in the group. You have helped me to bring the best out of myself. I also want to immensely thank my friends both at Purdue and at home for being my best stress busters. I would not have been here if it wasn't for my families endless love and support since my childhood. Thank you, Amma for all your sacrifices. Your strength will always drive me to become a better human being. Last but not the least, I am ever grateful to the School of Aeronautics and As- tronautics and NASA JSC NNX13AK60A grant for their exceptional facilities and financial support during my time at Purdue. v TABLE OF CONTENTS Page LIST OF TABLES :::::::::::::::::::::::::::::::::: vii LIST OF FIGURES ::::::::::::::::::::::::::::::::: viii ABSTRACT ::::::::::::::::::::::::::::::::::::: xii 1 INTRODUCTION :::::::::::::::::::::::::::::::: 1 1.1 Problem Motivation and Goals ::::::::::::::::::::::: 1 1.2 Previous Contributions ::::::::::::::::::::::::::: 3 1.3 Thesis Overview ::::::::::::::::::::::::::::::: 6 2 INTRODUCTION TO CIRCULAR RESTRICTED THREE BODY PROB- LEM :::::::::::::::::::::::::::::::::::::::: 9 2.1 The General n-Body Problem ::::::::::::::::::::::: 9 2.2 The General and Relative Three Body Problem ::::::::::::: 13 2.3 Circular Restricted Three Body Problem (CR3BP) ::::::::::: 15 2.4 Equations of Motion in the CR3BP :::::::::::::::::::: 15 2.5 Integrals of Motion - Jacobi Constant ::::::::::::::::::: 21 2.6 Libration Points - Equilibrium Solutions in the CR3BP ::::::::: 22 2.7 Forbidden Regions and Zero Velocity Surfaces :::::::::::::: 25 2.8 Coordinate Transformations :::::::::::::::::::::::: 29 2.9 Stability of Equilibrium Points ::::::::::::::::::::::: 31 3 NUMERICAL METHODS AND PERIODIC ORBITS :::::::::::: 41 3.1 State Transition Matrix :::::::::::::::::::::::::: 41 3.2 Differential Corrections ::::::::::::::::::::::::::: 44 3.3 Single Shooting or Simple Targeting Problem ::::::::::::::: 46 3.4 Multiple Shooting Algorithm :::::::::::::::::::::::: 48 3.5 Symmetric Properties in CR3BP ::::::::::::::::::::: 51 3.6 Symmetric Periodic Orbits ::::::::::::::::::::::::: 52 3.7 Continuation process :::::::::::::::::::::::::::: 55 3.7.1 Natural Parameter Continuation ::::::::::::::::: 56 3.7.2 Pseudo-arclength Continuation :::::::::::::::::: 57 3.8 Stability of Periodic Orbits ::::::::::::::::::::::::: 59 3.9 Bifurcations ::::::::::::::::::::::::::::::::: 65 3.10 3D Periodic Orbit Families ::::::::::::::::::::::::: 66 4 RESONANT ORBITS AND POINCARE´ MAPS ::::::::::::::: 71 4.1 Concept of Resonance ::::::::::::::::::::::::::: 71 vi Page 4.1.1 Resonances in Two-Body Model :::::::::::::::::: 73 4.1.2 Resonances in CR3BP ::::::::::::::::::::::: 77 4.2 Stability and Bifurcations of 2D Resonant Orbits :::::::::::: 82 4.3 Three Dimensional Resonances :::::::::::::::::::::: 87 4.4 Invariant Manifold Theory ::::::::::::::::::::::::: 88 4.4.1 Invariant Manifolds for Fixed points ::::::::::::::: 88 4.4.2 Poincar´eMaps ::::::::::::::::::::::::::: 93 4.4.3 Invariant Manifolds for Periodic Orbits :::::::::::::: 96 4.4.4 Computation of Manifolds for Unstable Resonant Orbits :::: 99 5 REFERENCE TRAJECTORY DESIGN IN EPHEMERIS MODEL :::: 104 5.1 Resonant - Near Rectilinear Halo Orbits :::::::::::::::: 104 5.2 Challenges in Ephemeris Trajectory Design ::::::::::::::: 108 5.2.1 Quasi-periodic Trajectories in Ephemeris Model :::::::: 109 5.2.2 Generating Nearby Reference Ephemeris Solutions ::::::: 117 5.3 New Stacking Sequences in CR3BP ::::::::::::::::::: 121 5.3.1 Alternative Stacking Sequence for Repeatable Behavior :::: 121 5.3.2 Periapsis Control - a Non-homogeneous Stacking Sequence :: 122 6 TRANSFER DESIGN INCORPORATING RESONANT ORBITS : METHOD- OLOGY AND RESULTS :::::::::::::::::::::::::::: 133 6.1 Representative NRHO - Departure Orbit :::::::::::::::: 133 6.2 Representative DRO - Arrival Orbit ::::::::::::::::::: 134 6.3 Theoritical Minimum Transfer Cost ::::::::::::::::::: 135 6.4 Arc Blending Scheme - Transfer Design Process :::::::::::: 139 6.5 Direct Transfers from NRHO to DRO :::::::::::::::::: 143 6.6 Transfers Incorporating Resonant Arcs ::::::::::::::::: 146 6.7 Transfers Incorporating Resonant Manifolds :::::::::::::: 148 6.8 Transfers Using Tangential Departure and Arrival Arcs :::::::: 156 6.9 Designing Locally Optimal Transfers :::::::::::::::::: 163 6.10 Higher-Fidelity Transfers from NRHO to DRO ::::::::::::: 171 7 SUMMARY ::::::::::::::::::::::::::::::::::: 182 7.1 Resonant Orbits and Manifolds in CR3BP ::::::::::::::: 182 7.2 Reference Ephemeris Trajectory Design Using Resonant Orbits :::: 183 7.3 Transfer Trajectory Design Using Resonant Orbits and Manifolds :: 184 7.4 Future Recommendations :::::::::::::::::::::::: 184 REFERENCES ::::::::::::::::::::::::::::::::::: 186 vii LIST OF TABLES Table Page 2.1 Characteristic quantities in the Earth-Moon and Sun-Earth system. ::: 18 2.2 Non-dimensional position coordinates of the libration points in the Earth- Moon and Sun-Earth systems ::::::::::::::::::::::::: 25 2.3 Value of Jacobi constant for libration points in the Earth-Moon and Sun- Earth systems. ::::::::::::::::::::::::::::::::: 26 3.1 Natural parameters of a representative L1 Lyapunov orbit. ::::::::: 57 4.1 Two-Body orbital parameters of Moon. :::::::::::::::::::: 73 6.1 9:2 L2 synodic NRHO parameters. ::::::::::::::::::::: 134 6.2 Representative DRO parameters. :::::::::::::::::::::: 134 6.3 Transfer using 1:2 resonant arc, Jacobi constant history. ::::::::: 147 6.4 Transfer using 1:2 resonant arc, transfer costs and TOF. ::::::::: 151 6.5 Transfer using 2:3 resonant arc, transfer costs and TOF. ::::::::: 151 6.6 Transfer using 3:4 resonant arc, transfer costs and TOF. ::::::::: 154 6.7 Transfer using 4:3 resonant arc, transfer costs and TOF. ::::::::: 154 6.8 Transfer using 3:4 unstable resonant manifold, transfer costs and TOF. : 157 6.9 Transfer type A, transfer costs and TOF. :::::::::::::::::: 162 6.10 Transfer type A, Jacobi constant history. :::::::::::::::::: 162 6.11 Transfer type B, transfer costs and TOF. :::::::::::::::::: 163 6.12 Transfer type B, Jacobi constant history. :::::::::::::::::: 163 6.13 ∆V and TOF comparison - fixed time optimal transfer - Transfer type A. 169 6.14 ∆V and TOF comparison - variable time optimal transfer - Transfer type A. ::::::::::::::::::::::::::::::::::::::: 169 6.15 ∆V and TOF comparison - fixed time optimal transfer -Transfer type B. 170 6.16 ∆V and TOF comparison - variable time optimal transfer - Transfer type B. ::::::::::::::::::::::::::::::::::::::: 170 viii LIST OF FIGURES Figure Page 2.1 Definitions in the n-body problem. :::::::::::::::::::::: 10 2.2 Definition for position of particle Pi relative to particle Pq. :::::::: 12 2.3 Definitions in the three-body problem. :::::::::::::::::::: 14 2.4 Definitions in the CR3BP. ::::::::::::::::::::::::::: 17 2.5 Equilibrium points in the rotating frame of CR3BP. :::::::::::: 24 2.6 ZVS and ZVC for (C > CL1 ) and (CL2 < C < CL1 ). ::::::::::::: 28 2.7 ZVS and ZVC for (CL3 < C < CL2 ), (CL4 < C < CL3 ) and (C < CL4 ). ::: 38 2.8 ZVC for (C = CL1 ), (C = CL2 ), ZVC (C = CL3 ) and ZVC (C = CL4 ). ::: 39 2.9 Orientation of rotating frame to the Inertial frame. ::::::::::::: 40 3.1 Illustration of Single shooting method. :::::::::::::::::::: 49 3.2 Illustration of Multiple shooting method. :::::::::::::::::: 51 3.3 A symmetric, periodic L1 Lyapunov orbit. :::::::::::::::::: 55 3.4 L1 Lyapunov family. :::::::::::::::::::::::::::::: 59 3.5 Periodic orbit stability regions. :::::::::::::::::::::::: 62 3.6 L1 Lyapunov family - In-plane Stability index ν1. :::::::::::::: 64 3.7 L1 Lyapunov family - Out-of-Plane stability index ν3. :::::::::::: 64 3.8 Types of Bifurcations. ::::::::::::::::::::::::::::: 66 3.9 L1 Lyapunov family - Halo family bifurcation. :::::::::::::::: 67 3.10 L1 Northern Halo family. :::::::::::::::::::::::::::: 69 3.11 L1 Northern Halo family projected onx ^-^z plane. ::::::::::::::: 69 4.1 Two-body 2:1 resonant orbit in the EM-inertial frame. ::::::::::: 76 4.2 Two-body 2:1 resonant orbit in the EM-rotating frame. ::::::::::: 76 4.3 2:1 Resonant orbit correction in the CR3BP. ::::::::::::::::: 81 4.4